Let $\{U_k\}$ be a sequence of independent random variables, with each variable being uniformly distributed over the interval $[0,2]$, and let $X_n = U_1 U_2\cdots U_n$ for $n \geq 1$.
(a) Determine in which of the senses (a.s., m.s., p., d.) the sequence $\{X_n\}$ converges as $n\to\infty$, and identify the limit, if any. Justify your answers.
(b) Determine the value of the constant $\theta$ so that the sequence $\{Y_n\}$ defined by $Y_n = n^\theta \ln(X_n)$
converges in distribution as $n\to\infty$ to a nonzero limit.
Best Answer
Let $T_k = \log U_k$ and $Y_n = \log X_n$. Then the $T_k$ are iid random variables with a particular distribution over the ray $(-\infty,\log 2]$ that can be worked out. You can now prove whatever you want about the convergence of $Y_n$ using what you know about sums of random variables; the answers can be translated back into statements about $X_n$.